Which of the following best describes the probability of liking 5 out of 13 tracks when playing a Spotify playlist with the random feature? A) 1/13 B) 5/13 C) 5/26 D) 5/91 E) 10/13
This question is asking for the probability of liking 5 out of 13 tracks when playing a Spotify playlist with the random feature.
The number of ways to choose 5 tracks from a playlist of 13 tracks is given by the combination formula: C(13, 5) = 13! / (5! * (13-5)!) = 13! / (5! * 8!) = (13 * 12 * 11 * 10 * 9) / (5 * 4 * 3 * 2 * 1) = 1287.
The total number of possible outcomes when randomly selecting 5 tracks from a playlist of 13 is given by the combination formula: C(13, 5) = 1287.
Therefore, the probability of liking 5 out of 13 tracks is 1/1287.
Thus, none of the given options accurately describe the probability. The correct answer should be "None of the above".
To determine the probability of liking 5 out of 13 tracks when playing a Spotify playlist with the random feature, we can use the concept of combinations.
The total number of possible outcomes is the number of ways to choose 5 tracks out of 13, which is denoted as 13C5 or "13 choose 5." This can be calculated using the formula:
13C5 = 13! / (5!(13-5)!)
where "!" denotes the factorial function.
The formula simplifies to:
13C5 = 13! / (5!8!)
Using this formula, we can calculate the value of 13C5:
13! = 13 * 12 * 11 * 10 * 9 * 8! = 95,040
5! = 5 * 4 * 3 * 2 * 1 = 120
8! = 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1 = 40,320
Plugging these values back into the formula:
13C5 = 95,040 / (120 * 40,320)
= 95,040 / 4,838,400
= 1 / 51
Therefore, the probability of liking 5 out of 13 tracks is 1/51. So, the correct answer is not one of the given options.
To find the probability of liking 5 out of 13 tracks when playing a Spotify playlist with the random feature, we need to use the concept of combinations.
In this case, we want to know the probability of selecting 5 tracks that you like out of the total 13 tracks. The total number of possible combinations when selecting 5 tracks out of 13 is given by the combination formula:
C(n, r) = n! / (r!(n - r)!)
where n is the total number of tracks (13 in this case) and r is the number of tracks you like (5 in this case).
C(13, 5) = 13! / (5!(13 - 5)!)
Simplifying the expression gives us:
C(13, 5) = 13! / (5!8!)
Calculating the factorial values:
13! = 13 x 12 x 11 x 10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1
5! = 5 x 4 x 3 x 2 x 1
8! = 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1
Plugging in the values:
C(13, 5) = (13 x 12 x 11 x 10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1) / ((5 x 4 x 3 x 2 x 1)(8 x 7 x 6 x 5 x 4 x 3 x 2 x 1))
Calculating the expression:
C(13, 5) = (128,648,725) / (120 x 40,320)
Simplifying the expression gives us:
C(13, 5) = (128,648,725) / (4,838,400)
Cancelling out common factors:
C(13, 5) = 26,334,449
Therefore, the probability of liking 5 out of 13 tracks is 26,334,449 out of the total possible combinations.
Now, let's check the options given:
A) 1/13: This option implies that there is a 1 in 13 chance of liking 5 tracks out of 13, which does not match the calculated probability.
B) 5/13: This option implies that there is a 5 in 13 chance of liking 5 tracks out of 13, which does not match the calculated probability.
C) 5/26: This option implies that there is a 5 in 26 chance of liking 5 tracks out of 13, which does not match the calculated probability.
D) 5/91: This option implies that there is a 5 in 91 chance of liking 5 tracks out of 13, which does not match the calculated probability.
E) 10/13: This option implies that there is a 10 in 13 chance of liking 5 tracks out of 13, which does not match the calculated probability.
Therefore, none of the given options match the calculated probability of 26,334,449 out of the total possible combinations.